Subgroups — #15756148

Order 3328.

350 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114967	C₂	1	1	yes	maximal, characteristic
2	#13113584	C₂	1	1	yes	maximal, characteristic
3	#13113400	C₂	1	2	yes	maximal
4	#13112696	C₂	1	2	yes	maximal
5	C₁₃×#2960	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#533	C₂²	1	1	yes	characteristic
7	C₁₃×#446	—	2	1	no	—
8	C₁₃×#446	C₄	1	2	yes	complemented
9	C₈×Q₈×C₁₃	—	2	1	no	—
10	C₈×Q₈×C₁₃	C₄	1	2	yes	complemented
11	C₁₃×#424	—	2	1	no	—
12	C₁₃×#423	C₂²	1	1	yes	characteristic
13	C₁₃×#405	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	#11718597	C₂²	1	2	yes	—
16	#11718586	C₂²	1	2	yes	—
17	#11718143	—	2	1	no	—
18	#11718143	—	2	1	no	—
19	#11718130	—	2	4	no	—
20	#11718050	—	2	1	no	—
21	#11718049	—	2	1	no	—
22	C₂×C₁₃×M₁₆	—	2	1	no	—
23	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₁₃×C₄⋊Q₈	Q₈	1	1	yes	characteristic
26	C₁₃×C₄⋊Q₈	Q₈	1	1	yes	characteristic
27	C₁₃×#125	—	2	1	no	—
28	C₁₃×C₂²⋊Q₈	—	2	1	no	—
29	C₁₃×C₂²⋊Q₈	D₄	1	2	yes	—
30	C₄×Q₈×C₁₃	—	2	1	no	—
31	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₁₃×#117	—	2	1	no	—
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₃	—	2	2	no	—
35	C₁₃×C₄⋊C₈	—	2	2	no	—
36	C₁₃×C₄⋊C₈	—	2	2	no	—
37	C₁₃×C₄⋊C₈	—	2	1	no	—
38	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₁₃×#97	—	2	2	no	—
40	C₄×C₈×C₁₃	—	2	2	no	—
41	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
45	#88071	—	2	2	no	—
46	#88071	—	2	2	no	—
47	C₂²⋊#1055	—	2	2	no	—
48	C₄⋊#1055	—	2	2	no	—
49	C₄×#1055	—	2	2	no	—
50	#8679	—	13	1	no	maximal, Sylow
51	C₂×Q₈×C₁₃	—	2	1	no	—
52	C₂×Q₈×C₁₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
56	C₂²×Dic₁₃	—	4	2	no	—
57	C₁₃×M₁₆	—	4	2	no	—
58	C₂×C₈×C₁₃	—	2	2	no	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₁₃×C₄⋊C₄	—	2	2	no	—
66	C₁₃×C₄⋊C₄	—	2	2	no	—
67	C₁₃×C₄⋊C₄	—	2	1	no	—
68	C₁₃×C₄⋊C₄	—	2	1	no	—
69	C₁₃×C₄⋊C₄	—	2	1	no	—
70	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
71	C₁₃×C₂²⋊C₄	—	2	2	no	—
72	C₄²×C₁₃	—	2	2	no	—
73	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
76	C₄²×C₁₃	#41	1	1	yes	characteristic
77	C₄⋊Dic₁₃	—	4	2	no	—
78	C₄⋊Dic₁₃	—	4	2	no	—
79	C₂×#1055	—	4	2	no	—
80	#2960	D₁₃	1	1	yes	characteristic
81	#1835	—	13	1	no	—
82	#1564	—	13	1	no	—
83	#1518	—	13	2	no	—
84	#1286	—	13	2	no	—
85	C₂³×C₁₃	#121	1	1	yes	characteristic
86	Q₈×C₁₃	—	4	2	no	—
87	Q₈×C₁₃	—	4	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	4	1	no	—
91	C₂×C₄×C₁₃	—	4	1	no	—
92	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#135	1	2	yes	—
99	C₂×C₄×C₁₃	#137	1	2	yes	—
100	C₂×Dic₁₃	—	8	2	no	—
101	C₂×Dic₁₃	—	8	2	no	—
102	C₂×Dic₁₃	—	8	2	no	—
103	C₈×C₁₃	—	2	1	no	—
104	C₈×C₁₃	—	2	1	no	—
105	C₈×C₁₃	—	2	1	no	—
106	C₈×C₁₃	—	2	1	no	—
107	C₈×C₁₃	—	4	2	no	—
108	#1055	—	8	2	no	—
109	#533	C₂×D₁₃	1	1	yes	characteristic
110	#446	—	2	1	no	—
111	#446	Dic₁₃	1	2	yes	complemented
112	C₈×Q₈	—	2	1	no	—
113	C₈×Q₈	Dic₁₃	1	2	yes	complemented
114	#431	—	13	2	no	—
115	#424	—	2	1	no	—
116	#423	C₂×D₁₃	1	1	yes	characteristic
117	#405	—	2	1	no	—
118	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
119	#382	—	13	2	no	—
120	#340	—	26	1	no	—
121	#340	—	26	1	no	—
122	#335	—	26	1	no	—
123	#334	—	26	1	no	—
124	#328	—	26	4	no	—
125	C₂²×C₁₃	—	2	1	no	—
126	C₂²×C₁₃	—	2	1	no	—
127	C₂²×C₁₃	#380	1	1	yes	characteristic
128	C₂²×C₁₃	#464	1	2	yes	—
129	C₄×C₁₃	—	2	1	no	—
130	C₄×C₁₃	—	2	1	no	—
131	C₄×C₁₃	—	2	1	no	—
132	C₄×C₁₃	—	2	1	no	—
133	C₄×C₁₃	—	4	2	no	—
134	C₄×C₁₃	—	4	2	no	—
135	C₄×C₁₃	#416	1	1	yes	characteristic
136	C₄×C₁₃	#419	1	1	yes	characteristic
137	C₄×C₁₃	#479	1	2	yes	—
138	Dic₁₃	—	16	2	no	—
139	C₂×M₁₆	—	2	1	no	—
140	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
141	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
142	C₄⋊Q₈	Dic₂₆	1	1	yes	characteristic
143	C₄⋊Q₈	Dic₂₆	1	1	yes	characteristic
144	#125	—	2	1	no	—
145	C₂²⋊Q₈	—	2	1	no	—
146	C₂²⋊Q₈	D₅₂	1	2	yes	—
147	C₄×Q₈	—	2	1	no	—
148	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
149	#117	—	2	1	no	—
150	C₂×C₄⋊C₄	—	26	2	no	—
151	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	1	no	—
155	C₄⋊C₈	—	26	2	no	—
156	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
157	C₂²⋊C₈	—	26	2	no	—
158	#97	—	2	2	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	26	2	no	—
161	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
162	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
165	#95	—	26	2	no	—
166	#95	—	26	2	no	—
167	C₂×C₁₃	—	2	2	no	—
168	C₂×C₁₃	#1805	1	1	yes	characteristic
169	C₂×C₁₃	#1812	1	1	yes	characteristic
170	C₂×C₁₃	#1814	1	1	yes	characteristic
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	C₄⋊Dic₁₃	1	2	yes	—
173	C₂²×C₄	—	52	2	no	—
174	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
175	C₂²×C₄	#6127	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
177	M₁₆	—	4	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	52	2	no	—
182	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
183	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
184	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
185	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	52	2	no	—
192	C₄⋊C₄	—	52	2	no	—
193	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
197	C₄²	#6127	1	1	yes	characteristic
198	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
199	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
200	C₁₃	#8679	1	1	yes	characteristic, complemented, Sylow
201	C₂³	#88184	1	1	yes	characteristic
202	Q₈	—	4	2	no	—
203	Q₈	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	104	2	no	—
209	C₂×C₄	—	104	2	no	—
210	C₂×C₄	—	104	2	no	—
211	C₂×C₄	#88178	1	1	yes	characteristic
212	C₂×C₄	#88178	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
214	C₂×C₄	#88180	1	1	yes	characteristic
215	C₂×C₄	#88184	1	1	yes	characteristic
216	C₂×C₄	#88196	1	2	yes	—
217	C₂×C₄	#88204	1	2	yes	—
218	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	4	2	no	—
224	C₈	—	104	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	#11718689	1	2	yes	—
228	C₂²	#11718873	1	1	yes	characteristic
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	208	2	no	—
236	C₄	#11718683	1	2	yes	—
237	C₄	#11718886	1	1	yes	characteristic
238	C₄	#11718894	1	1	yes	characteristic
239	C₂	—	2	2	no	—
240	C₂	#13114851	1	1	yes	characteristic
241	C₂	#13114882	1	1	yes	characteristic
242	C₂	#13114890	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718873
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2960	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂³×C₁₃	#121
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#2960	D₁₃
hypocenter	C₁₃	#8679
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
8. C₁₃×#446	C₄	semidirect
10. C₈×Q₈×C₁₃	C₄	semidirect
111. #446	Dic₁₃	semidirect
113. C₈×Q₈	Dic₁₃	semidirect
200. C₁₃	#8679	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13114967	2
#13113584	2
#13113400	2
#13112696	2
C₁₃×#2960	2
#8679	13